| Abstract: |
| We present some recent advances in the productive and symbiotic interplay between general potential theories (subharmonic functions associated to closed subsets $\mathcal{F} \subset \mathcal{J}^2(X)$ of the 2-jets on $X \subset \mathbb{R}^n$ open) and subsolutions of degenerate elliptic PDEs of the form $F(x,u,Du,D^2u) = 0$. We will describe the {\em monotonicity-duality} method begin by Harvey and Lawson [Comm.\ Pure Appl.\ Math, 2009] for proving comparison principles for potential theories where $\mathcal{F}$ has {\em sufficient monotonicity} and {\em fiberegularity} (in variable coefficient settings) and which carry over to all differential operators $F$ which are {\em compatible} with $\mathcal{F}$ in a precise sense.
Particular attention will be given to {\em gradient dependent} examples with the requisite sufficient monotonicity of {\em proper ellipticity} and {\em directionality}. Examples operators we will discuss include those of {\em optimal transport} in which the target density is strictly increasing in some directions as well as operators which are parabolic in the sense of Krylov. Further examples, modeled on {\em hyperbolic polynomials} in the sense of G\aa rding, produce additional examples in which the comparison principles holds, but standard viscosity structural conditions fail to hold. |
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